Shape restrictions such as monotonicity, convexity, and concavity provide a natural way of limiting the complexity of many statistical estimation problems. Shape-constrained estimation is not as well understood as more traditional nonparametric estimation involving smoothness constraints. For instance, the minimax rate of convergence for multivariate convex regression has yet to be rigorously established in full generality. Even the one-dimensional case is challenging, and has been of recent interest (Guntuboyina and Sen, 2013). In this paper we study the problem of variable selection in multivariate convex regression. Assuming that the regression function is convex and sparse, our goal is to identify the relevant variables. We show that it suffices to estimate a sum of onedimensional convex functions, leading to significant computational and statistical advantages. This is in contrast to general nonparametric regression, where fitting an additive model can result in false negatives. Our approach is based on a twostage quadratic programming procedure.

Stochastic variational inference (SVI) lets us scale up Bayesian computation to massive data. It uses stochastic optimization to fit a variational distribution, following easy-to-compute noisy natural gradients. As with most traditional stochastic optimization methods, SVI takes precautions to use unbiased stochastic gradients whose expectations are equal to the true gradients. In this paper, we explore the idea of following biased stochastic gradients in SVI. Our method replaces the natural gradient with a similarly constructed vector that uses a fixed-window moving average of some of its previous terms. We will demonstrate the many advantages of this technique. First, its computational cost is the same as for SVI and storage requirements only multiply by a constant factor. Second, it enjoys significant variance reduction over the unbiased estimates, smaller bias than averaged gradients, and leads to smaller mean-squared error against the full gradient. We test our method on latent Dirichlet allocation with three large corpora.

Bell's Theorem shows that quantum mechanical correlations can violate the constraints that the causal structure of certain experiments impose on any classical explanation. It is thus natural to ask to which degree the causal assumptions -- e.g. locality or measurement independence -- have to be relaxed in order to allow for a classical description of such experiments. Here, we develop a conceptual and computational framework for treating this problem. We employ the language of Bayesian networks to systematically construct alternative causal structures and bound the degree of relaxation using quantitative measures that originate from the mathematical theory of causality. The main technical insight is that the resulting problems can often be expressed as computationally tractable linear programs. We demonstrate the versatility of the framework by applying it to a variety of scenarios, ranging from relaxations of the measurement independence, locality and bilocality assumptions, to a novel causal interpretation of CHSH inequality violations.

This paper studies large-scale dynamical networks where the current state of the system is a linear transformation of the previous state, contaminated by a multivariate Gaussian noise. Examples include stock markets, human brains and gene regulatory networks. We introduce a transition matrix to describe the evolution, which can be translated to a directed Granger transition graph, and use the concentration matrix of the Gaussian noise to capture the second-order relations between nodes, which can be translated to an undirected conditional dependence graph. We propose regularizing the two graphs jointly in topology identification and dynamics estimation. Based on the notion of joint association graph (JAG), we develop a joint graphical screening and estimation (JGSE) framework for efficient network learning in big data. In particular, our method can pre-determine and remove unnecessary edges based on the joint graphical structure, referred to as JAG screening, and can decompose a large network into smaller subnetworks in a robust manner, referred to as JAG decomposition. JAG screening and decomposition can reduce the problem size and search space for fine estimation at a later stage. Experiments on both synthetic data and real-world applications show the effectiveness of the proposed framework in large-scale network topology identification and dynamics estimation.

We consider classification tasks in the regime of scarce labeled training data in high dimensional feature space, where specific expert knowledge is also available. We propose a new hybrid optimization algorithm that solves the elastic-net support vector machine (SVM) through an alternating direction method of multipliers in the first phase, followed by an interior-point method for the classical SVM in the second phase. Both SVM formulations are adapted to knowledge incorporation. Our proposed algorithm addresses the challenges of automatic feature selection, high optimization accuracy, and algorithmic flexibility for taking advantage of prior knowledge. We demonstrate the effectiveness and efficiency of our algorithm and compare it with existing methods on a collection of synthetic and real-world data.

The Fisher information matrix (FIM) is a foundational concept in statistical signal processing. The FIM depends on the probability distribution, assumed to belong to a smooth parametric family. Traditional approaches to estimating the FIM require estimating the probability distribution function (PDF), or its parameters, along with its gradient or Hessian. However, in many practical situations the PDF of the data is not known but the statistician has access to an observation sample for any parameter value. Here we propose a method of estimating the FIM directly from sampled data that does not require knowledge of the underlying PDF. The method is based on non-parametric estimation of an $f$-divergence over a local neighborhood of the parameter space and a relation between curvature of the $f$-divergence and the FIM. Thus we obtain an empirical estimator of the FIM that does not require density estimation and is asymptotically consistent. We empirically evaluate the validity of our approach using two experiments.

We present a simple but powerful reinterpretation of kernelized locality-sensitive hashing (KLSH), a general and popular method developed in the vision community for performing approximate nearest-neighbor searches in an arbitrary reproducing kernel Hilbert space (RKHS). Our new perspective is based on viewing the steps of the KLSH algorithm in an appropriately projected space, and has several key theoretical and practical benefits. First, it eliminates the problematic conceptual difficulties that are present in the existing motivation of KLSH. Second, it yields the first formal retrieval performance bounds for KLSH. Third, our analysis reveals two techniques for boosting the empirical performance of KLSH. We evaluate these extensions on several large-scale benchmark image retrieval data sets, and show that our analysis leads to improved recall performance of at least 12%, and sometimes much higher, over the standard KLSH method.

Graph connection Laplacian (GCL) is a modern data analysis technique that is starting to be applied for the analysis of high dimensional and massive datasets. Motivated by this technique, we study matrices that are akin to the ones appearing in the null case of GCL, i.e the case where there is no structure in the dataset under investigation. Developing this understanding is important in making sense of the output of the algorithms based on GCL. We hence develop a theory explaining the behavior of the spectral distribution of a large class of random matrices, in particular random matrices with random block entries of fixed size. Part of the theory covers the case where there is significant dependence between the blocks. Numerical work shows that the agreement between our theoretical predictions and numerical simulations is generally very good.

Random ferns is a machine learning algorithm proposed by [11] for matching same elements between two images of the same scene, allowing one to recognise certain objects or trace them on videos. The original motivation behind this method was to create a simple and efficient algorithm by extending the Naïve Bayes classifier; still the authors acknowledged its strong connection to the decision tree ensembles like the Random forest [2] algorithm. Since introduction, Random ferns have been applied in numerous computer vision application, like image recognition [1], action recognition [10] or augmented reality [14]. However, it has not gathered attention outside this field; thus, this work aims to bring this algorithm to a much wider spectrum of applications. In order to do that, I propose a generalised version of the algorithm, implemented as an R [13] package rFerns. The paper is organised as follows. Section 2 briefly recalls the Bayesian derivation of the original version of Random ferns, presents the decision tree ensemble interpretation of the algorithm and lists modifications leading to the rFerns variant.

Synaptic plasticity is believed to be the fundamental building block of learning and memory in the brain. Its study is of crucial importance to understanding the activity and function of neural circuits. With innovations in neural recording technology providing access to the simultaneous activity of increasingly large populations of neurons, statistical models are promising tools for formulating and testing hypotheses about the dynamics of synaptic connectivity. Advances in optical techniques (Packer et al., 2012; Hochbaum et al., 2014), for example, have made it possible to simultaneously record from and stimulate large populations of synaptically connected neurons. Armed with statistical tools capable of inferring time-varying synaptic connectivity, neuroscientists could test competing models of synaptic plasticity, discover new learning rules at the monosynaptic and network level, investigate the effects of disease on synaptic plasticity, and potentially design stimuli to modify neural networks. Despite the popularity of GLMs for spike data, relatively little work has attempted to model the time-varying nature of neural interactions. Here we model interaction weights as a dynamical system governed by parametric synaptic plasticity rules. To perform inference in this model, we use particle Markov Chain Monte Carlo (pMCMC) (Andrieu et al., 2010), a recently developed inference technique for complex time series. We use this new modeling framework to examine the problem of using recorded data to distinguish between proposed variants of spike-timing-dependent plasticity (STDP) learning rules.